$\int \frac{1}{\sqrt{4-9 x^2}} \mathit{\ⅆ}x$

$\= \int \frac{\frac{2}{3}\cos \left(\mathrm{\θ}\right)}{2 \cos \left(\mathrm{\θ}\right)} \mathit{\ⅆ}\mathrm{\θ}$

$\= \int \frac{1}{3} \mathit{\ⅆ}\mathrm{\θ}$

$\=\mathrm{\θ}\/3$

$\=\frac{1}{3} \arcsin \left(\frac{3}{2}x\right)$

Evaluate the given integral

$$$\int \frac{1}{\sqrt{4-9 x^2}} \mathit{\ⅆ}x$ = $\frac{1}{3}\arcsin \left(\frac{3}{2}x\right)$$$

Initialization

$\mathrm{with}\left(\mathrm{IntegrationTools}\right)\:$

$$$Q≔\int \frac{1}{\sqrt{4-9 x^2}} \mathit{\ⅆ}x\:$

Change variables as per Table 6.3.1

$q_1≔\mathrm{Change}\left(Q\,x\=\frac{2}{3}\sin \left(\mathrm{\θ}\right)\right)$

$\∫\left(-\frac{1}{6}\frac{\sqrt{-4{\sin \left(\mathrm{\θ}\right)}^2\+4}\cos \left(\mathrm{\θ}\right)}{{\sin \left(\mathrm{\θ}\right)}^2-1}\right)\ⅆ\mathrm{\θ}$

$q_2≔\mathrm{simplify}\left(q_1\right) assuming \mathrm{\θ}\colon\colon \mathrm{RealRange}\left(-\frac{\mathrm{\π}}{2}\,\frac{\mathrm{\π}}{2}\right)$

$\frac{1}{3}\∫1\ⅆ\mathrm{\θ}$

$q_3≔\mathrm{value}\left(q_2\right)$

$\frac{1}{3}\mathrm{\θ}$

$\mathrm{eval}\left(q_3\,\mathrm{\θ}\=\arcsin \left(\frac{3}{2}x\right)\right)$

$\frac{1}{3}\arcsin \left(\frac{3}{2}x\right)$

$\begin{array}{cccc}\multicolumn{4}{c}{\int \frac{1}{\sqrt{-9x^2+4}}\ⅆx}\\ & \text{=}& \int \frac{1}{3}\ⅆu\hfill & \left[\mathrm{change}\,x=\frac{2\sin \left(u\right)}{3}\,u\right]\\ & \text{=}& \frac{u}{3}\hfill & \left[\mathrm{constant}\right]\\ & \text{=}& \frac{\arcsin \left(\frac{3x}{2}\right)}{3}\hfill & \left[\mathrm{revert}\right]\end{array}$

Table 6.3.2(a)   Annotated stepwise solution via Integration Methods tutor